For new problems consider a cubic block of wood that is
composed of 27 smaller cubes in a 3x3x3 arrangement.
The first
problem concerns a worm that is in the center cube. It can
tunnel through any of the six faces of the cube
to an adjacent
cube, but it cannot tunnel diagonally to another cube. Is it
possible for the worm, moving in this
way from cube to cube, to
visit every one of the 27 cubes in the larger 3x3x3 cube exactly
once and then return
to the center cube that he started at? If
your answer is yes provide a path. If your answer is no, you
should explain why not.
A second problem involving the 3x3x3 cube is the following.
Suppose one wishes to cut the cube into its 27 smaller
cubes
using straight planar cuts. One way to do this is to use six
cuts, two in each of the three dimensions of the
cube. Suppose
that one is allowed to rearrange the pieces of the cube after
each cut so that the next cut will slice
several pieces at once.
Is it possible to dissect the cube into its 27 smaller cubes in
fewer than six cuts if one is
allowed these rearrangements? Once
again either provide the method or explain why it can't be done
with fewer
than six cuts. This problem is also worth
investigating for a 4x4x4 cube.